Mathematics of a Lady Tasting Tea
Mathematics 15: Lecture 19
Dan Sloughter
Furman University
November 2, 2006
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
1/9
Ronald Aylmer Fisher
I
1890 - 1962
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
2/9
Ronald Aylmer Fisher
I
1890 - 1962
I
Two classic books on statistics: Statistical Methods for Research
Workers, first published in 1925, and The Design of Experiments, first
published in 1935
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
2/9
Ronald Aylmer Fisher
I
1890 - 1962
I
Two classic books on statistics: Statistical Methods for Research
Workers, first published in 1925, and The Design of Experiments, first
published in 1935
I
Equally famous as a geneticist (for example, the text The Genetical
Theory of Natural Selection)
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
2/9
Ronald Aylmer Fisher
I
1890 - 1962
I
Two classic books on statistics: Statistical Methods for Research
Workers, first published in 1925, and The Design of Experiments, first
published in 1935
I
Equally famous as a geneticist (for example, the text The Genetical
Theory of Natural Selection)
I
Refused a prestigious position in London to pursue statistical
problems in agriculture at Rothamsted, where he developed, among
many other fundamental notions of modern statistics, the theory of
randomized experimental design
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
2/9
The experiment
I
Given a cup of tea with milk, a lady claims she can discriminate as to
whether milk or tea was first added to the cup.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
3/9
The experiment
I
Given a cup of tea with milk, a lady claims she can discriminate as to
whether milk or tea was first added to the cup.
I
To test her claim, eight cups of tea are prepared, four of which have
the milk added first and four of which have the tea added first.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
3/9
The experiment
I
Given a cup of tea with milk, a lady claims she can discriminate as to
whether milk or tea was first added to the cup.
I
To test her claim, eight cups of tea are prepared, four of which have
the milk added first and four of which have the tea added first.
I
Question: How many cups does she have to correctly identify to
convince us of her ability?
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
3/9
Choosing subsets
I
There are 8 × 7 × 6 × 5 = 1680 ways to choose a first cup, a second
cup, a third cup, and a fourth cup, in order.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
4/9
Choosing subsets
I
There are 8 × 7 × 6 × 5 = 1680 ways to choose a first cup, a second
cup, a third cup, and a fourth cup, in order.
I
There are 4 × 3 × 2 × 1 = 24 ways to order four cups.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
4/9
Choosing subsets
I
There are 8 × 7 × 6 × 5 = 1680 ways to choose a first cup, a second
cup, a third cup, and a fourth cup, in order.
I
There are 4 × 3 × 2 × 1 = 24 ways to order four cups.
I
So the number of ways to choose 4 cups out of 8 is
1680
= 70.
24
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
4/9
Choosing subsets
I
There are 8 × 7 × 6 × 5 = 1680 ways to choose a first cup, a second
cup, a third cup, and a fourth cup, in order.
I
There are 4 × 3 × 2 × 1 = 24 ways to order four cups.
I
So the number of ways to choose 4 cups out of 8 is
1680
= 70.
24
I
Note: the lady performs the experiment by selecting 4 cups, say, the
ones she claims to have had the tea poured first.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
4/9
Choosing subsets
I
There are 8 × 7 × 6 × 5 = 1680 ways to choose a first cup, a second
cup, a third cup, and a fourth cup, in order.
I
There are 4 × 3 × 2 × 1 = 24 ways to order four cups.
I
So the number of ways to choose 4 cups out of 8 is
1680
= 70.
24
I
Note: the lady performs the experiment by selecting 4 cups, say, the
ones she claims to have had the tea poured first.
I
For example, the probability that she would correctly identify all 4
1
cups is 70
.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
4/9
Choosing 3
I
To get exactly 3 right, and, hence, 1 wrong, she would first have to
choose 3 from the 4 correct ones.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
5/9
Choosing 3
I
To get exactly 3 right, and, hence, 1 wrong, she would first have to
choose 3 from the 4 correct ones.
I
She can do this 4 × 3 × 2 = 24 ways with order.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
5/9
Choosing 3
I
To get exactly 3 right, and, hence, 1 wrong, she would first have to
choose 3 from the 4 correct ones.
I
I
She can do this 4 × 3 × 2 = 24 ways with order.
Since 3 cups can be ordered in 3 × 2 = 6 ways, there are 4 ways for her
to choose the 3 correctly.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
5/9
Choosing 3
I
To get exactly 3 right, and, hence, 1 wrong, she would first have to
choose 3 from the 4 correct ones.
I
I
I
She can do this 4 × 3 × 2 = 24 ways with order.
Since 3 cups can be ordered in 3 × 2 = 6 ways, there are 4 ways for her
to choose the 3 correctly.
Since she can now choose the 1 incorrect cup 4 ways, there are a total
of 4 × 4 = 16 ways for her to choose exactly 3 right and 1 wrong.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
5/9
Choosing 3
I
To get exactly 3 right, and, hence, 1 wrong, she would first have to
choose 3 from the 4 correct ones.
I
I
She can do this 4 × 3 × 2 = 24 ways with order.
Since 3 cups can be ordered in 3 × 2 = 6 ways, there are 4 ways for her
to choose the 3 correctly.
I
Since she can now choose the 1 incorrect cup 4 ways, there are a total
of 4 × 4 = 16 ways for her to choose exactly 3 right and 1 wrong.
I
Hence the probability that she chooses exactly 3 correctly is
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
16
70
=
November 2, 2006
8
35 .
5/9
Statistical significance
I
Suppose the lady correctly identifies all 4 cups.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
6/9
Statistical significance
I
I
Suppose the lady correctly identifies all 4 cups.
Conclusion
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
6/9
Statistical significance
I
I
Suppose the lady correctly identifies all 4 cups.
Conclusion
I
Either she has no ability, and has chosen the correct 4 cups purely by
chance, or
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
6/9
Statistical significance
I
I
Suppose the lady correctly identifies all 4 cups.
Conclusion
I
I
Either she has no ability, and has chosen the correct 4 cups purely by
chance, or
she has the discriminatory ability she claims.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
6/9
Statistical significance
I
I
Suppose the lady correctly identifies all 4 cups.
Conclusion
I
I
I
Either she has no ability, and has chosen the correct 4 cups purely by
chance, or
she has the discriminatory ability she claims.
Since choosing correctly is highly unlikely in the first case (one chance
in seventy), we decide for the second.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
6/9
Statistical significance
I
I
Suppose the lady correctly identifies all 4 cups.
Conclusion
I
I
Either she has no ability, and has chosen the correct 4 cups purely by
chance, or
she has the discriminatory ability she claims.
I
Since choosing correctly is highly unlikely in the first case (one chance
in seventy), we decide for the second.
I
Note: if she got 3 correct and 1 wrong, this would be evidence for her
ability, but not persuasive evidence since the chance of getting 3 or
17
more correct is 70
= 0.2429.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
6/9
Statistical significance
I
I
Suppose the lady correctly identifies all 4 cups.
Conclusion
I
I
Either she has no ability, and has chosen the correct 4 cups purely by
chance, or
she has the discriminatory ability she claims.
I
Since choosing correctly is highly unlikely in the first case (one chance
in seventy), we decide for the second.
I
Note: if she got 3 correct and 1 wrong, this would be evidence for her
ability, but not persuasive evidence since the chance of getting 3 or
17
more correct is 70
= 0.2429.
I
Note: typically, a result is considered statistically significant if the
probability of its occurrence is less than 0.05, that is, less than 1 out
of 20.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
6/9
The null hypothesis
I
The null hypothesis is a specific description of a possible state of
nature.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
7/9
The null hypothesis
I
The null hypothesis is a specific description of a possible state of
nature.
I
In this example, the null hypothesis is the hypothesis that the lady
has no special ability to discriminate between the cups of tea.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
7/9
The null hypothesis
I
The null hypothesis is a specific description of a possible state of
nature.
I
In this example, the null hypothesis is the hypothesis that the lady
has no special ability to discriminate between the cups of tea.
I
Note: we can never prove the null hypothesis, but the data may
provide evidence to reject it.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
7/9
The null hypothesis
I
The null hypothesis is a specific description of a possible state of
nature.
I
In this example, the null hypothesis is the hypothesis that the lady
has no special ability to discriminate between the cups of tea.
I
Note: we can never prove the null hypothesis, but the data may
provide evidence to reject it.
I
Note: in most situations, rejecting the null hypothesis is what we
hope to do.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
7/9
Randomization
I
It is randomization which allows us to make the probability
calculations which reveal whether the data are significant or not.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
8/9
Randomization
I
It is randomization which allows us to make the probability
calculations which reveal whether the data are significant or not.
I
Randomization takes care of all the possible causes for which we
cannot control.
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
8/9
Problems
1. Suppose the lady samples 10 cups of tea, among which 5 had the tea
poured first and 5 had the milk poured first.
a. What is the probability she correctly identifies all five cups which had the
tea poured first?
b. What is the probability she correctly identifies exactly four of the cups
which had the tea poured first?
c. What is the probability she correctly identifies four or more of the cups
which had the tea poured first?
d. Would we reject the null hypothesis if she correctly identified exactly four
of the cups which had the tea poured first?
Dan Sloughter (Furman University)
Mathematics of a Lady Tasting Tea
November 2, 2006
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